Reformulated Severi dimensionality conjecture for unicuspidal rational curves

Let ${\rm S}$ be a numerical semigroup with conductor $c$, and let ${\bf k}=(k_1,\dots,k_n)$ be a ramification profile. Let $B$ be the set of Betti elements of the ramification orders $k_i$, together with the minimal generators of ${\rm S}$ less than $c$. For $s\in {\rm S}$, let $\rho(s)$ be the number of elements of $\mathbb{N}\setminus {\rm S}$ strictly larger than $s$. Let $\phi(s)$ count the factorizations contributed by the Betti element $s$ that do not arise from strictly smaller Betti elements, and let ${\bf k}^{\bullet}$ be the distinguished proper subset of minimal generators less than $c$ that are not in ${\bf k}$. For positive integers $d$, $g$, and $n$ with $n\leq 2g\leq d$, suppose that the Severi variety $\mathcal{V}_{\bf k}\subset M^n_{d,g}$ is nonempty. Reformulated Severi dimensionality conjecture.

$${\rm cod}(\mathcal{V}_{\bf k},M^n_d)=\sum_{i=1}^n(k_i-i)+\sum_{s\in B}\phi(s)\rho(s)-\sum_{s\in{\bf k}^{\bullet}}\rho(s)-1.$$

This adjusts the earlier Severi dimensionality conjecture by expressing the expected codimension through Betti elements and their factorizations. The conjecture concerns the codimension of Severi varieties of unicuspidal rational curves and is presented as a general statement beyond the hyperelliptic and supersymmetric cases proved in the paper.